Shapes of implied volatility with positive mass at zero

We study the shapes of the implied volatility when the underlying distribution has an atom at zero. We show that the behaviour at small strikes is uniquely determined by the mass of the atom at least up to the third asymptotic order, regardless of the properties of the remaining (absolutely continuous, or singular) distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can—a priori—distinguish between mass at the origin and a heavy-left-tailed distribution. An atom at zero is found in stochastic models with absorption at the boundary, such as the CEV process, and can be used to model default events, as in the class of jump-to-default structural models of credit risk. We numerically test our model-free result in such examples. Note that while Lee’s moment formula [21] tells that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as [2, 14] do not apply in this setting essentially due to the breakdown of Put-Call symmetry—and we rely here on an alternative treatment of the problem.